On generators of polynomial algebras in two commuting or non-commuting variables

Abstract

An element of a free associative algebra A 2 = K〈 x 1, x 2〉 is called primitive if it is an automorphic image of x 1. We address the problem of detecting primitive elements of A 2: we present an algorithm that distinguishes primitive elements, and also give a couple of very handy necessary conditions for primitivity that allow one to rule out many sorts of non-primitive elements of A 2 just by inspection. We also give a structural description of the automorphism groups Aut( A 2) and Aut( P 2) (where P 2 = K[ x 1, x 2] is the polynomial algebra in two variables over the same ground field K) which is different from previously known descriptions.